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Regularity of a boundary having a Schwarz function. (English) Zbl 0728.30007

Summary: In his book, P. J. Davis discussed various interesting aspects concerning a Schwarz function. It is a holomorpic function S defined in a neighbourhood of a real analytic arc satisfying \(S(\zeta)={\bar \zeta}\) on the arc, where \({\bar \zeta}\) denotes the complex conjugate of \(\zeta\). The author defines a Schwarz function for a portion of a boundary of an arbitrary open set and shows the regularity of the portion of the boundary. More precisely, let \(\Omega\) be an open set of the unit disk B such that the boundary \(\partial \Omega\) contains the origin 0 and let \(\Gamma =(\partial \Omega)\cap B\). A function S defined on \(\Omega\cup \Gamma\) is called the Schwarz function of \(\Omega\cup \Gamma\) if (i) S is holomorphic in \(\Omega\), (ii) S is continuous on \(\Omega\cup \Gamma\) and (iii) \(S(\zeta)={\bar \zeta}\) on \(\Gamma\). The author gives a classification of a boundary having a Schwarz function. The main theorem asserts that there are four types of the boundary if 0 is not an isolated boundary point of \(\Omega\) : 0 is a regular, nonisolated degenerate, double or cusp point of the boundary.

MSC:

30C35 General theory of conformal mappings
30C99 Geometric function theory
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[1] Ahlfors, L. V. &Sario, L.,Riemann Surfaces. Princeton Univ. Press, Princeton, 1960.
[2] Caffarelli, L. A. &Rivière, N. M., Smoothness and analyticity of free boundaries in variational inequalities.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 289–310. · Zbl 0363.35009
[3] – Asymptotic behaviour of free boundaries at their singular points.Ann. of Math. (2), 106 (1977), 309–317. · Zbl 0364.35041
[4] Collingwood, E. F. &Lohwater, A. J.,The Theory of Cluster Sets. Cambridge Univ. Press, Cambridge, 1966. · Zbl 0149.03003
[5] Davis, P. J.,The Schwarz Function, and its Applications. Carus Math. Monographs, No. 17. Math. Assoc. America, Washington, D.C., 1974. · Zbl 0293.30001
[6] Fuchs, W. H. J., A Phragmén-Lindelöf theorem conjectured, by D. J. Newman.Trans. Amer. Math. Soc., 267 (1981), 285–293. · Zbl 0472.30025
[7] Nevanlinna, R.,Eindeutige analytische Funktionen. 2nd ed., Springer, Berlin, 1953.
[8] Sario, L. &Nakai, M.,Classification Theory of Riemann Surfaces. Springer, Berlin, 1970. · Zbl 0199.40603
[9] Schaeffer, D. G., Some examples of singularities in a free boundary.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4 (1977), 133–144. · Zbl 0354.35033
[10] Shapiro, H. S., Unbounded quadrature domains. InComplex Analysis Vol. I Lecture Notes in Math. Vol. 1275. Springer, Berlin, 1987, pp. 287–331. · Zbl 0634.30037
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